Diophantine approximation on manifolds dates back to (1932) the profound conjecture of K. Mahler [1]}, which can be rephrased as the statement that \(^1(_n(\Psi )\cap _n)=0\)  for \(\Psi (q)=q^{-\tau }\)  with \(\tau >n\) . Here and throughout \(_n=(x, x^2, \ldots , x^n)\)  denotes the Veronese curve. Mahler's conjecture was eventually proven in 1965 by Sprindžuk [2]}, who then conjectured that the same holds when \(_n\)  is replaced by any nondegenerate analytic manifold, and \(^1\)  is replaced by \(^k\)  where \(k\)  is the dimension of this manifold. Although particular cases of Sprindžuk's conjecture were known, it was not until 1998 that Kleinbock &
Margulis [3]} established Sprindžuk's conjecture in full generality by using dynamical tools based on diagonal flows on homogeneous spaces. This breakthrough result acted as a catalyst for the subsequent progress in this area of research. The convergence Lebesgue measure result for \(_n(\Psi )\cap \)  was established first in [4]} and then in a generalised form in [5]}. The divergence case was established for the Veronese curve in [6]} and for any non-degenerate manifolds in [7]}.
The first inhomogeneous Lebesgue measure result for the divergence case of \(_n(\Psi )\cap \)  was established very recently in [8]}.
